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Theorem pweqb 4223
Description: Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
Assertion
Ref Expression
pweqb (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)

Proof of Theorem pweqb
StepHypRef Expression
1 sspwb 4216 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 sspwb 4216 . . 3 (𝐵𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
31, 2anbi12i 460 . 2 ((𝐴𝐵𝐵𝐴) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
4 eqss 3170 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3170 . 2 (𝒫 𝐴 = 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
63, 4, 53bitr4i 212 1 (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wss 3129  𝒫 cpw 3575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598
This theorem is referenced by: (None)
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